ex-7: reword efficiency paragraph

notes/sections/7.md

@@ -289,41 +289,40 @@ samples. Plots in @fig:percep_proj.
 
 ## Efficiency test
 
-A program was implemented in order to check the validity of the two
-aforementioned methods.  
-A number $N_t$ of test samples was generated and the
-points were divided into the two classes according to the selected method.
-At each iteration, false positives and negatives are recorded using a running
-statistics method implemented in the `gsl_rstat` library, being suitable for
-handling large datasets for which it is inconvenient to store in memory all at
-once.  
-For each sample, the numbers $N_{fn}$ and $N_{fp}$ of false positive and false
-negative are computed with the following trick: every noise point $x_n$ was
-checked this way: the function $f(x_n)$ was computed with the weight vector $w$
-and the $t_{\text{cut}}$ given by the employed method, then:
-      
+A program was implemented to check the validity of the two
+classification methods.  
+A number $N_t$ of test samples, with the same parameters of the training set,
+is generated using an RNG and their points are divided into noise/signal by
+both methods. At each iteration, false positives and negatives are recorded
+using a running statistics method implemented in the `gsl_rstat` library, to
+avoid storing large datasets in memory.  
+In each sample, the numbers $N_{fn}$ and $N_{fp}$ of false positive and false
+negative are obtained in this way: for every noise point $x_n$ compute the
+activation function $f(x_n)$ with the weight vector $w$ and the
+$t_{\text{cut}}$, then:
+
   - if $f(x) < 0 \thus$ $N_{fn} \to N_{fn}$
   - if $f(x) > 0 \thus$ $N_{fn} \to N_{fn} + 1$ 
 
-Similarly for the positive points.  
-Finally, the mean and the standard deviation were computed from $N_{fn}$ and
-$N_{fp}$ obtained for every sample in order to get the mean purity $\alpha$
-and efficiency $\beta$ for the employed statistics:
+and similarly for the positive points.  
+Finally, the mean and standard deviation are computed from $N_{fn}$ and
+$N_{fp}$ of every sample and used to estimate purity $\alpha$
+and efficiency $\beta$ of the classification:
 
 $$
   \alpha = 1 - \frac{\text{mean}(N_{fn})}{N_s} \et
   \beta = 1 - \frac{\text{mean}(N_{fp})}{N_n}
 $$
 
-Results for $N_t = 500$ are shown in @tbl:res_comp. As can be observed, the
-Fisher method gives a nearly perfect assignment of the points to their belonging
-class, with a symmetric distribution of false negative and false positive,
-whereas the points perceptron-divided show a little more false-positive than
-false-negative, being also more changable from dataset to dataset.  
-The reason why this happened lies in the fact that the Fisher linear
-discriminant is an exact analitical result, whereas the perceptron is based on
-a convergent behaviour which cannot be exactely reached by definition.
-
+Results for $N_t = 500$ are shown in @tbl:res_comp. As can be seen, the
+Fisher discriminant gives a nearly perfect classification
+with a symmetric distribution of false negative and false positive,
+whereas the perceptron show a little more false-positive than
+false-negative, being also more variable from dataset to dataset.  
+A possible explanation of this fact is that, for linearly separable and
+normally distributed points, the Fisher linear discriminant is an exact
+analytical solution, whereas the perceptron is only expected to converge to the
+solution and thus more subjected to random fluctuations.
 
 
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